Sharp freezing time estimates for the subcritical Facilitated Exclusion Process

Abstract

We investigate the exact transience time of the Facilitated Exclusion Process (FEP) on the one-dimensional torus with $N$ sites. The FEP exhibits an active/inactive phase transition at critical density $1/2$, such that in the subcritical density regime $(0,1/2)$, it becomes frozen after a finite time period -- the transience time or freezing time. We first show that for the FEP starting from a Bernoulli product measure of marginal density $\ddot{I}\in (0,1/2)$, the transience time has exactly the scale of $\hat{I}˜({\log }^{3}N)$. Secondly, we prove that in the near-critical case $\ddot{I}\simeq 1/2-N^{-\hat{I}\pm }$ for $\hat{I}\pm \in (0,1)$, the transience time is polynomial and has a scale of $N^{1\wedge (2\hat{I}\pm )}$. The key idea is to estimate the typical size of locally supercritical intervals of the initial distribution, which has order $\log N$ in the subcritical case and $N^{1\wedge (2\hat{I}\pm )}$ in the near-critical case. In the subcritical case this is enough, whereas in the near-critical case we need additional dynamical decorrelation inequalities to apply this static result to estimate the freezing time.

34 pages, 3 figures. Comments welcome!